CK12 - Geometry

In this lessonwe will investigatepropertiesof thesesegmentsand solvea varietyof problems.

Propertiesof midsegmentswithintriangles

We start with a theoremthat we will use to solveproblemsthat involvemidsegmentsof triangles.

MidsegmentTheorem:The segmentthat joins the midpointsof a pair of sidesof a triangle

is:

1. parallelto the third side.


2. half as long as the third side.

Proofof 1. We needto showthat a midsegmentis parallelto the third side.We will do this usingthe Parallel
Postulate.

Considerthe followingtriangle. Constructthe midpoint of side.

By the ParallelPostulate,thereis exactlyone line though that is parallelto side. Let’s say that it

intersectsside at point. We will showthat must be the midpointof and then we can conclude

that is a midsegmentof the triangleand is parallelto.

We mustshowthat the line through and parallelto side will intersectside at its midpoint.If
a parallelline cuts off congruentsegmentson one transversal,then it cuts off congruentsegmentson every

transversal.This ensuresthat point is the midpointof side.

Since , we have. Hence,by the definitionof midpoint,point is the midpoint

of side. is a midsegmentof the triangleand is also parallelto.

Proofof 2. We mustshowthat.

In , constructthe midpointof side at point and midsegments and as follows: